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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
NASA NTRS2014

Bifurcation Analysis Using Rigorous Branch and Bound Methods

For the study of nonlinear dynamic systems, it is important to locate the equilibria and bifurcations occurring within a specified computational domain. This paper proposes a new approach for solving these problems and compares it to the numerical continuation method. The new approach is based upon branch and bound and utilizes rigorous enclosure techniques to yield outer bounding sets of both the equilibrium and local bifurcation manifolds. These sets, which comprise the union of hyper-rectangles, can be made to be as tight as desired. Sufficient conditions for the existence of equilibrium and bifurcation points taking the form of algebraic inequality constraints in the state-parameter space are used to calculate their enclosures directly. The enclosures for the bifurcation sets can be computed independently of the equilibrium manifold, and are guaranteed to contain all solutions within the computational domain. A further advantage of this method is the ability to compute a near-maximally sized hyper-rectangle of high dimension centered at a fixed parameter-state point whose elements are guaranteed to exclude all bifurcation points. This hyper-rectangle, which requires a global description of the bifurcation manifold within the computational domain, cannot be obtained otherwise. A test case, based on the dynamics of a UAV subject to uncertain center of gravity location, is used to illustrate the efficacy of the method by comparing it with numerical continuation and to evaluate its computational complexity.

Smith, Andrew P.↗
NASA NTRS1975

A two parameter survey of periodic orbits in the restricted problem of three bodies

Within the context of the restricted problem of three bodies the effects caused by varying the mass ratio of the primaries and the eccentricity of their orbits, upon periodic orbits of the infinitesimal mass which are numerical continuations of circular orbits in the ordinary problem of two bodies are shown. A recursive power series technique is used to numerically integrate the equations of motion as well as the first variational equations in order to generate a two parameter family of perodic orbits and identify the linear stability characteristics. Seven such families are investigated with equally spaced mass ratios from 0.0 to 1.0 and eccentricities of the orbits of the primaries in a range 0.0 to 0.6. Stable orbits are associated with large distances of the infinitesimal mass from the perturbing primary, nearly circular motion of the primaries, and small mass ratios of the primaries. Unstable orbits for the infinitesimal mass are associated with small distances from the perturbing primary, highly elliptic orbits of the primaries and large mass ratios.

Shelus, P. J.↗
NASA NTRS1990

Bifurcations and sensitivity in parametric nonlinear programming

The parametric nonlinear programming problem is that of determining the behavior of solution(s) as a parameter or vector of parameters alpha belonging to R(sup r) varies over a region of interest for the problem: Minimize over x the set f(x, alpha):h(x, alpha) = 0, g(x, alpha) is greater than or equal to 0, where f:R(sup (n+r)) approaches R, h:R(sup (n+r)) approaches R(sup q) and g:R(sup (n+r)) approaches R(sup p) are assumed to be at least twice continuously differentiable. Some of these parameters may be fixed but not known precisely and others may be varied to enhance the performance of the system. In both cases a fundamentally important problem in the investigation of global sensitivity of the system is to determine the stability boundaries of the regions in parameter space which define regions of qualitatively similar solutions. The objective is to explain how numerical continuation and bifurcation techniques can be used to investigate the parametric nonlinear programming problem in a global sense. Thus, first the problem is converted to a closed system of parameterized nonlinear equations whose solution set contains all local minimizers of the original problem. This system, which will be represented as F(z,alpha) = O, will include all Karush-Kuhn-Tucker and Fritz John points, both feasible and infeasible solutions, and relative minima, maxima, and saddle points of the problem. The local existence and uniqueness of a solution path (z(alpha), alpha) of this system as well as the solution type persist as long as a singularity in the Jacobian D(sub z)F(z,alpha) is not encountered. Thus the nonsingularity of this Jacobian is characterized in terms of conditions on the problem itself. Then, a class of efficient predictor-corrector continuation procedures for tracing solution paths of the system F(z,alpha) = O which are tailored specifically to the parametric programming problem are described. Finally, these procedures and the obtained information are illustrated within the context of design optimization.

Lundberg, Bruce N.↗
NASA NTRS1978

Flow field visualization about external axial corners

An experimental investigation was conducted to visualize the flow field about external axial corners. The investigation was initiated to provide answers to questions about the inviscid flow pattern for continuing numerical investigations. Symmetrical and asymmetrical corner models were tested at a Reynolds number per meter of 60,700,000. Oil-flow and vapor-screen photographs were taken for both models at angle of attack and yaw. The paper presents the results of the investigation in the form of oil-flow photographs and the surrounding shock wave location obtained from the vapor screens.

Talcott, N. A., Jr.↗
NASA NTRS1982

Periodic orbits of the elliptic restricted problem for the Sun-Jupiter-Saturn system

A systematic approach to generate periodic orbits in the elliptic restricted problem of three bodies is introduced. The approach is based on (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to periodic orbits in the elliptic restricted problem. Two families of periodic orbits of the elliptic restricted problem are found by this approach. The mass ratio of the primaries of these orbits is equal to that of the Sun-Jupiter system. The sidereal mean motions between the infinitesimal body and the smaller primary are in a 2:5 resonance, so as to approximate the Sun-Jupiter-Saturn system. The lineaar stabilities of these periodic orbits are studied as functions of the eccentricities of the primaries and of the infinitesimal body. The results show that both stable and unstable periodic orbits exist in the elliptic restricted problem that are close to the actual Sun-Jupiter-Saturn system. However, the periodic orbit closest to the actual Sun-Jupiter-Saturn system is (linearly) stable.

Kwok, J. H.↗
NASA NTRS1985

Periodic orbits of the general three-body problem for the sun-Jupiter-Saturn system

Two families of symmetric periodic orbits of the planar, general, three-body problem are presented. The masses of the three bodies include ratios equal to the sun-Jupiter-Saturn system and the periods of the orbits of Jupiter and Saturn are in a 2:5 resonance. The (linear) stability of the orbits are studied in relation to eccentricity and mass variations. The generation of the two families of periodic orbits follows a systematic approach and employs (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to the elliptic restricted problem and from the circular and elliptic problems to the general problem through bifurcation phenomena relating the three dynamical systems. The approach also provides insight into the evolutionary process of periodic orbits continued from the restricted problems to the general problem.

Kwok, J. H.↗
NASA NTRS1993

Multi-layered reasoning by means of conceptual fuzzy sets

The real world consists of a very large number of instances of events and continuous numeric values. On the other hand, people represent and process their knowledge in terms of abstracted concepts derived from generalization of these instances and numeric values. Logic based paradigms for knowledge representation use symbolic processing both for concept representation and inference. Their underlying assumption is that a concept can be defined precisely. However, as this assumption hardly holds for natural concepts, it follows that symbolic processing cannot deal with such concepts. Thus symbolic processing has essential problems from a practical point of view of applications in the real world. In contrast, fuzzy set theory can be viewed as a stronger and more practical notation than formal, logic based theories because it supports both symbolic processing and numeric processing, connecting the logic based world and the real world. In this paper, we propose multi-layered reasoning by using conceptual fuzzy sets (CFS). The general characteristics of CFS are discussed along with upper layer supervision and context dependent processing.

Takagi, Tomohiro↗
NASA NTRS2006

Uncertainty Analysis for a Jet Flap Airfoil

An analysis of variance (ANOVA) study was performed to quantify the potential uncertainties of lift and pitching moment coefficient calculations from a computational fluid dynamics code, relative to an experiment, for a jet flap airfoil configuration. Uncertainties due to a number of factors including grid density, angle of attack and jet flap blowing coefficient were examined. The ANOVA software produced a numerical model of the input coefficient data, as functions of the selected factors, to a user-specified order (linear, 2-factor interference, quadratic, or cubic). Residuals between the model and actual data were also produced at each of the input conditions, and uncertainty confidence intervals (in the form of Least Significant Differences or LSD) for experimental, computational, and combined experimental / computational data sets were computed. The LSD bars indicate the smallest resolvable differences in the functional values (lift or pitching moment coefficient) attributable solely to changes in independent variable, given just the input data points from selected data sets. The software also provided a collection of diagnostics which evaluate the suitability of the input data set for use within the ANOVA process, and which examine the behavior of the resultant data, possibly suggesting transformations which should be applied to the data to reduce the LSD. The results illustrate some of the key features of, and results from, the uncertainty analysis studies, including the use of both numerical (continuous) and categorical (discrete) factors, the effects of the number and range of the input data points, and the effects of the number of factors considered simultaneously.

Green, Lawrence L.↗
NASA NTRS1989

Experimental and numerical investigation of an oblique shock wave/turbulent boundary layer interaction with continuous suction

An numerical and experimental investigation has been conducted into the interaction of an incident oblique shock wave with a turbulent boundary layer, for the cases of a rough plate and a porous plate with suction, at a nominal Mach number of 2.5 and flow deflection angles of 0, 4, 6, and 8 deg. Attention is given to the pitot pressure profiles, wall static pressures, and porous plate local bleed distributions measured for the two plates. Suction is found to increase the strength of the incident shock required to separate the boundary layer; for all shock strengths tested, separation is completely eliminated.

Benhachmi, Driss↗
NASA NTRS2015

Numerical Computation of a Continuous-thrust State Transition Matrix Incorporating Accurate Hardware and Ephemeris Models

A significant body of work exists showing that providing a nonlinear programming (NLP) solver with expressions for the problem constraint gradient substantially increases the speed of program execution and can also improve the robustness of convergence, especially for local optimizers. Calculation of these derivatives is often accomplished through the computation of spacecraft's state transition matrix (STM). If the two-body gravitational model is employed as is often done in the context of preliminary design, closed form expressions for these derivatives may be provided. If a high fidelity dynamics model, that might include perturbing forces such as the gravitational effect from multiple third bodies and solar radiation pressure is used then these STM's must be computed numerically. We present a method for the power hardward model and a full ephemeris model. An adaptive-step embedded eight order Dormand-Prince numerical integrator is discussed and a method for the computation of the time of flight derivatives in this framework is presented. The use of these numerically calculated derivatieves offer a substantial improvement over finite differencing in the context of a global optimizer. Specifically the inclusion of these STM's into the low thrust missiondesign tool chain in use at NASA Goddard Spaceflight Center allows for an increased preliminary mission design cadence.

Low Thrust↗
NASA NTRS1999

Recent Progress on the VASIMR

Experimental and theoretical studies on the Variable Specific Impulse Magnetoplasma Rocket (VASIMR) have continued through a NASA led collaborative program involving several research groups. In the experimental area, performance characterization of the VASIMR helicon plasma source has been obtained over a portion of the parameter space, with helium and hydrogen propellant. Density (10(exp 18) - 10(exp 19)/ cubic meter) and temperature (5 eV) were measured at moderate degree of ionization in two separate experimental devices. Helicon design improvement and optimization will be discussed. Experiments with the ion cyclotron resonance heating (ICRH) subsection have begun and preliminary results will be discussed. Theoretical picture and integrated numerical simulation continue to be refined to account for the main physics elements of the VASIMR, including RF absorption and particle acceleration with subsequent detachment in the magnetic nozzle.

ChangDiaz, Franklin R.↗
NASA NTRS1997

Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains

Borrowing from techniques developed for conservation law equations, numerical schemes which discretize the Hamilton-Jacobi (H-J), level set, and Eikonal equations on triangulated domains are presented. The first scheme is a provably monotone discretization for certain forms of the H-J equations. Unfortunately, the basic scheme lacks proper Lipschitz continuity of the numerical Hamiltonian. By employing a virtual edge flipping technique, Lipschitz continuity of the numerical flux is restored on acute triangulations. Next, schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous Hamiltonians. These schemes possess a discrete maximum principle on arbitrary triangulations and naturally exhibit proper Lipschitz continuity of the numerical Hamiltonian. Finally, a class of Petrov-Galerkin approximations are considered. These schemes are stabilized via a least-squares bilinear form. The Petrov-Galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy.

Barth, Timothy J.↗
NASA NTRS1974

Optimal output feedback control of linear systems in presence of forcing and measurement noise

The problem of obtaining an optimal control law, which is constrained to be a linear feedback of the available measurements, for both continuous and discrete time linear systems subjected to additive white process noise and measurement noise was Necessary conditions are obtained for minimizing a quadratic performance function for both finite and infinite terminal time cases. The feedback gains are constrained to be time invariant for the infinite terminal time cases. For all the cases considered, algorithms are derived for generating sequences of feedback gain matrices which successively improve the performance function. A continuous time numerical example is included for the purpose of demonstration.

Joshi, S. M.↗
NASA NTRS1987

A finite difference scheme for three-dimensional steady laminar incompressible flow

A finite difference scheme for three-dimensional steady laminar incompressible flows is presented. The Navier-Stokes equations are expressed conservatively in terms of velocity and pressure increments (delta form). First order upwind differences are used for first order partial derivatives of velocity increments resulting in a diagonally dominant matrix system. Central differences are applied to all other terms for second order accuracy. The SIMPLE pressure correction algorithm is used to satisfy the continuity equation. Numerical results are presented for cubic cavity flow problems for Reynolds numbers up to 2000 and are in good agreement with other numerical results.

Hwang, Danny P.↗
NASA NTRS1987

A finite difference scheme for three-dimensional steady laminar incompressible flow

A finite difference scheme for three-dimensional steady laminar incompressible flows is presented. The Navier-Stokes equations are expressed conservatively in terms of velocity and pressure increments (delta form). First order upwind differences are used for first order partial derivatives of velocity increments resulting in a diagonally dominant matrix system. Central differences are applied to all other terms for second order accuracy. The SIMPLE pressure correction algorithm is used to satisfy the continuity equation. Numerical results are presented for cubic cavity flow problems for Reynolds numbers up to 2000 and are in good agreement with other numerical results.

Hwang, Danny P.↗